We present a new numerical scheme to solve the Helmholtz equation in awave-guide. We consider a medium that is bounded in the $x_2$-direction,unbounded in the $x_1$-direction and $\varepsilon$-periodic for large $|x_1|$,allowing different media on the left and on the right. We suggest a newnumerical method that is based on a truncation of the domain and the use ofBloch wave ansatz functions in radiation boxes. We prove the existence and astability estimate for the infinite dimensional version of the proposedproblem. The scheme is tested on several interfaces of homogeneous and periodicmedia and it is used to investigate the effect of negative refraction at theinterface of a photonic crystal with a positive effective refractive index.
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